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NUMERICAL ASSESSMENT OF REVERSE-FLOW MUFFLERS USING ASIMULATED ANNEALING METHOD
Min-Chie ChiuDepartment of Automatic Control Engineering, Chungchou Institute of Technology, Yuanlin, Changhua 51003, R.O.C.
E-mail: minchie.chiu@msa.hinet.net
Received December 2008, Accepted March 2010
No. 08-CSME-45, E.I.C. Accession 3117
ABSTRACT
Because of the necessity of maintenance and operation in industries in which the equipment layoutis occasionally tight, the space for a muffler is constrained. An interest in maximizing the acousticalperformance of mufflers within a limited space is of paramount importance. As mufflers hybridizedwith reverse-flow ducts may visibly increase acoustical performance, the main purpose of this paperis to numerically analyze and maximize their acoustical performance within a limited space. In thispaper, a four-pole system matrix for evaluating the acoustic performance —sound transmission loss(STL)— is derived by using a decoupled numerical method. Moreover, simulated annealing (SA), arobust scheme used to search for the global optimum by imitating the metal annealing process, hasbeen used during the optimization process. Before dealing with a broadband noise, the STL’smaximization with respect to a one-tone noise (300 Hz) is introduced for a reliability check on the SAmethod. Moreover, an accuracy check of the mathematical model is performed. Results reveal thatthe STL of a muffler with reverse-flow perforated ducts can be maximized at the desired frequencyfor pure tone elimination; moreover, the noise reduction for a broadband noise can reach 97.5 dB.Consequently, the approach used for the optimal design of the mufflers is simple and effective.
METHODE DU RECUIT SIMULE POUR L’EVALUATION NUMERIQUE DEL’ECOULEMENT INVERSE D’UN SILENCIEUX
RESUME
Dans l’industrie ou la configuration de l’espace pour l’appareillage est parfois limitee, et a causede la necessite de faire de la maintenance, ainsi que pour l’operation de l’appareillage, l’espace pourun silencieux est restreint. L’interet de maximiser la performance acoustique des silencieux al’interieur d’un espace limite est d’une importance capitale. Etant donne que des silencieux hybridesa conduits a ecoulement inverse peuvent assurement augmenter la performance acoustique, le butprincipal de l’article est l’analyse numerique et la maximisation de leur performance acoustique dansun espace limite. Dans notre recherche, un systeme matricielle a quatre poles pour l’evaluation de laperformance acoustique—perte de transmission sonore (STL) — est derive en utilisant une methodea commande numerique decouple. De plus, la methode du recuit simule (SA), un schema robusteutilise pour la recherche d’un optimum global en imitant le procede technique du recuit simule enmetallurgie, a ete utilisee durant le processus d’optimisation. Avant de s’interesser au bruit a largebande, la maximisation de la perte de transmission sonore, pour un bruit d’un ton pur (300 Hz) estintroduit pour verifier la fiabilite de la methode du recuit simule (SA). En outre, une verification del’exactitude du modele mathematique est executee. Les resultats demontrent que la perte detransmission sonore d’un silencieux avec ecoulement inverse et conduits perfores peuvent etremaximises a la frequence desiree pour l’elimination de bruit d’un ton pur; de plus la reduction dubruit pour un bruit a large bande peut atteindre 97.5 dB. Par consequent, l’approche utilisee pour ledesign optimal du silencieux est simple et efficace.
Transactions of the Canadian Society for Mechanical Engineering, Vol. 34, No. 1, 2010 17
1. INTRODUCTION
Research on mufflers was started by Davis et al. in 1954 [1]. To overcome the exhaust noise of aventing system, the assessment of new perforated-element mufflers was started by Sullivan andCrocker in 1978 [2]. Based on the coupled equations derived by Sullivan and Crocker in 1978 [2], aseries of theory and numerical techniques in decoupling the acoustical problems have beenproposed [3–7]; however, the restrictions of non-flow and instability problems in the solution stillexisted; fortunately, Munjal [8] and Peat [9] promulgated the generalized decoupling andnumerical decoupling methods in 1987 and 1988 which overcome the drawbacks in the previous
NOMENCLATURE
Co: sound speed (m s-1)c1, c2: coefficientsdhi: the diameter of a perforated
hole on the i-th inner tube(m )
D: diameter of the tubes (m)f: cyclic frequency (Hz)Iter: maximum iterationj: imaginary unitk: wave number (5
v
co
)
kk: cooling rate in SAkf1, kf2,
kf3, kf4,
kf5, kf6: coefficients in function CCi5
kfiellix
L1,L2: lengths of inlet/outlet straightducts (m)
Lo: total length of the muffler (m)M: mean flow Mach numberOBJi: objective function (dB)p: acoustic pressure (Pa)�ppi: acoustic pressure at the i-th
node (Pa)pb Tð Þ: transition probabilityQ: volume flow rate of venting
gas (m3 s21)Si: section area at the i-th node(m2)STL: sound transmission loss (dB)SWLO: unsilenced sound power level
inside the muffler’s inlet (dB)SWLT : overall sound power level
inside the muffler’s output(dB)
ti: the thickness of the ith innerperforated tube (m)
TS1ij,TS3ij: components of four-pole
transfer matrices for anacoustical mechanism withstraight ducts
TPRF2ij: components of a four-poletransfer matrix for an acous-tical mechanism with reverse-flow perforated ducts
T�ij: components of a four-poletransfer system matrix
u: acoustic particle velocity(m s21)
�uui: acoustic particle velocity atthe i-th node (m s21)
ui,j: acoustical particle velocitypassing through a perforatedhole from the ith node to thejth node (m s21)
Vi: mean flow velocity at the ithnode (m s21)
ro: air density (kg m23)ri: acoustical density at the ith
nodeji: specific acoustical impedance
of the ith inner perforatedtube
gi: the porosity of the ith innerperforated tube.
li: ith eigen value of LL½ �6x6
PP½ �6x6: the model matrix formed byan eigen vector PP6x1 ofLL½ �6x6
Transactions of the Canadian Society for Mechanical Engineering, Vol. 34, No. 1, 2010 18
studies. To increase a muffler’s acoustical performance, the assessment of a new acousticalelement — a reverse-flow mechanism with double internal perforated tubes — was proposed andinvestigated by Munjal et al. in 1987 [8]. Due to the necessity of operation and maintenance withinan enclosed machine room, a space-constrained problem within a noise abatement facility occurs;therefore, there is a growing need to optimize the acoustical performance within a fixed space.Yet, the need to investigate the optimal muffler design under space constraints is rarely addressed.In previous papers, the shape optimizations of simple-expansion mufflers were discussed [10–13].To greatly enlarge the acoustical performance within a fixed space, the acoustical mechanism ofthe above mufflers hybridized with reverse-flow perforated tubes using a novel scheme ofsimulated annealing (SA) is presented.
2. THEORETICAL BACKGROUND
In this paper, a muffler with reverse-flow perforated mufflers was adopted for noiseelimination in the blower room shown in Fig. 1. The outlines of these mufflers are shown inFig. 2. Before the acoustical fields of mufflers are analyzed, the acoustical elements have to bedistinguished. As shown in Fig. 1, two kinds of muffler components, including two straightducts and a reverse-flow perforated duct, are identified and symbolized as I and II. In addition,the acoustic pressure �pp and acoustic particle velocity �uu within the muffler are depicted in Fig. 2where the acoustical field is represented by four nodes (node 1, 2, 4, and 5).
The muffler system is composed of two kinds of acoustical elements. The individual transfermatrix derivations with respect to two kinds of acoustical mechanisms are described below.
Fig. 1. A distinction in a muffler with reverse-flow ducts installed in a space-constrained blowerroom.
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2.1. System MatrixBased on plane wave theory, the four-pole transfer matrix for a straight duct between nodes 1
and 2 is [14]
�pp1
roco�uu1
� �~e
{jM1kL1
1{M21
TS11,1 TS11,2
TS12,1 TS12,2
� ��pp2
roco�uu2
� �ð1aÞ
where
TS11,1~cosk(L1zLA)
1{M21
� �; TS11,2~j sin
k(L1zLA)
1{M21
� �; TS12,1~j sin
k(L1zLA)
1{M21
� �;
TS12,2~cosk(L1zLA)
1{M21
� � ð1bÞ
Similarly, the relationship between nodes 4 and 5 is
�pp4
roco�uu4
� �~e
{jM4k(L1zLA)
1{M24
TS31,1 TS31,2
TS32,1 TS32,2
� ��pp5
roco�uu5
� �ð2aÞ
where
TS31,1~cosk L1zLAð Þ
1{M24
� �; TS31,2~j sin
k L1zLAð Þ1{M2
4
� �; TS32,1~j sin
k L1zLAð Þ1{M2
4
� �;
TS32,2~cosk L1zLAð Þ
1{M24
� � ð2bÞ
As derived in Appendix A, on the basis of plane wave theory, the four-pole transfer matrixfor a reverse-flow perforated duct between nodes 2 and 4 is
Fig. 2. The outline and acoustical field of a muffler with reverse-flow ducts.
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�pp2
roco�uu2
� �~
TPRF21,1 TPRF21,2
TPRF22,1 TPRF22,2
� ��pp4
roco�uu4
� �ð3Þ
The total transfer matrix assembled by multiplication from Eqs. (1),(3) is
�pp1
roco�uu1
!
~e{jk
M1 L1zLAð Þ1{M2
1
zM4 L1zLAð Þ
1{M24
� �TS11,1 TS11,2
TS12,1 TS12,2
" #TPFR21,1 TPRF21,2
TPRF22,1 TPRF22,2
" #
TS31,1 TS31,2
TS32,1 TS32,2
" #�pp5
roco�uu5
!ð4Þ
A simplified form in the matrix is expressed as
�pp1
roco�uu1
� �~
T�11 T�12
T�21 T�22
� ��pp5
roco�uu5
� �ð5Þ
Under the assumption of a fixed thickness of the tubes (t15t250.001 m) and the symmetricdesign (LA5LB5(LZ-LC)/2), the sound transmission loss (STL) of a muffler is definedas [14]
STL Q, f , RT1, RT2, RT3, RT4, dh1, g1, dh2, g2ð Þ
~logT�11zT�12zT�21zT�22
�� ��2
� �z10 log
S1
S5
� � ð6aÞ
where
RT1~ LZ=Lo; RT2~ LC=LZ; RT3~ D1=Do; RT4~ D2=Do;
Lo~L1zLZ; LZ~LAzLBzLC; LA~LB~ LZ{LCð Þ=2ð6bÞ
2.2. Overall Sound Power LevelThe silenced octave sound power level emitted from a silencer’s outlet is
SWLi~SWLOi{STLi ð7Þ
where (1) SWLOi is the original SWL at the inlet of a muffler (or pipe outlet), and i is the indexof the octave band frequency.
(2) STLi is the muffler’s STL with respect to the relative octave band frequency.
(3) SWLi is the silenced SWL at the outlet of a muffler with respect to the relative octaveband frequency.
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Finally, the overall SWLT silenced by a muffler at the outlet is
SWLT~10 � logX6
i~1
10SWLi=10
( )
~10 � log
10SWLO f ~125ð Þ{½
STL f ~125ð Þ�=10 z10½SWLO f ~250ð Þ{STL f ~250ð Þ�=10 z10
SWLO f ~500ð Þ{½STL f ~500ð Þ�=10
z10SWLO f ~1000ð Þ{½
STL f ~1000ð Þ�=10 z10SWLO f ~2000ð Þ{½
STL f ~2000ð Þ�=10
z10 SWL f ~4000ð Þ{STL f ~4000ð Þ½ �=10
8>>>><>>>>:
9>>>>=>>>>;
ð8Þ
2.3. Objective FunctionBy using the formulas of Eqs. (6) and (8), the objective function used in the SA optimization
was established.
STL Maximization for a One-tone (f) Noise:
OBJ1~STL Q,f ,RT1,RT2,RT3,RT4,dh1,g1,dh2,g2ð Þ ð9Þ
SWL Minimization for a Broadband Noise:To minimize the overall SWLT, the objective function is
OBJ2~SWLT Q,RT1,RT2,RT3,RT4,dh1,g1,dh2,g2ð Þ ð10Þ
3. MODEL CHECK
Before performing the SA optimal simulation on mufflers, an accuracy check of themathematical model on a muffler with reverse-flow perforated tubes was performed by Munjalet al. [8]. As indicated in Fig. 3, the accuracy comparisons between theoretical data and
Fig. 3. Performance of a one-chamber reverse-flow perforated muffler [D150.0493(m), D150.0493(m),Do50.1481(m), LA5LB50.0064, Lc50.1286(m), t15t250.0081(m), dh15dh250.0035(m), g15g250.039,M150.1], [Analytical data is from Munjal et al. [8]].
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analytical data are in agreement. Therefore, the mathematic model of mufflers with reverse-flowand perforated tubes is acceptable and adopted in the following optimization process.
4. CASE STUDIES
In this paper, the noise reduction of a space-constrained blower is exemplified and shown inFig. 1. The sound power level (SWL) inside the blower’s outlet is shown in Table 1 where theoverall SWL reaches 140.7 dB. To depress the huge exhaust noise emitted from the blower’soutlet, a muffler hybridized with reverse-flow tubes installed under the ground is considered.
To obtain the best acoustical performance within a fixed space volume, numerical assessmentslinked to a SA optimizer are applied. Before the minimization of a broadband noise is executed, areliability check of the SA method by maximization of the STL at one targeted tone (300 Hz) hasbeen carried out. As shown in Figs. 1 and 2, the available space for a muffler is 0.6 m in width,0.6 m in height, and 1.2 m in length. The flow rate (Q) and thickness of a perforated tube (t) arepreset as 0.05 (m3/s) and 0.001(m), respectively. The corresponding OBJ functions, spaceconstraints, and the ranges of design parameters are summarized in Table 2.
5. SIMULATED ANNEALING
The basic concept behind SA was first introduced by Metropolis et al. [15] and developed byKirkpatrick et al. [16]. SA simulates the annealing of metal. Annealing is the process of heatingand keeping a metal at a stabilized temperature while cooling it slowly. Slow cooling allows theparticles to attain their state close to the minimal energy state. The algorithm starts bygenerating a random initial solution. The scheme of the SA is a variation of the hill-climbingalgorithm. All downhill movements for improvement are accepted for the decrement of thesystem’s energy. The purpose of SA is to avoid stacking in local optimal solutions duringoptimization. In order to escape from the local optimum, the SA also allows movementsresulting in solutions that are worse (uphill moves) than the current solution. To imitate theevolution of the SA algorithm, a new random solution (X’) is chosen from the neighborhood ofthe current solution (X). If the change in objective function (or energy) is negative (DFƒ0), thenew solution will be accepted as the new current solution with the transition property pb(X’) 5
1. If the change is not negative (DFw0), the new transition property pb(X’) will be computedby the Boltzmann’s factor pb(X’) 5 exp DF=CTð Þ as the Eq. (11)
Table 1. The spectrum of exhaust sound power levels (SWLs).
f(Hz) 125 250 500 1k 2k 4k OverallSWLO-dB 130 140 128 115 110 105 140.7
Table 2. Range of design parameters for a muffler with reverse-flow ducts.
Range of design parameters
A muffler withreverse-flow ducts
Targeted f 5300 (Hz); Q50.05 (m3/s); Lo51.2 (m); Do50.6 (m)RT1:[0.5, 0.9]; RT2:[0.1, 0.9]; RT3:[0.1, 0.4]; RT4:[0.1, 0.4]; g1:[0. 03, 0.1]; dh1:[0. 00175, 0.007]; g2:[0. 03, 0.1]; dh2:[0. 00175,0.007];
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pb X 0ð Þ~1,DFƒ0
exp{DF
CT
� �,Df w0
0@ ð11aÞ
DF~F X 0ð Þ{F Xð Þ ð11bÞwhere C and T are the Boltzmann constant and the current temperature. Moreover, comparedwith the new random probability of rand(0,1), each successful substitution of the new currentsolution will lead to the decay of the current temperature as
Tnew ~ kk � Told
where kk is the cooling rate. The process is repeated until the predetermined number (Iter) ofthe outer loop is reached.
The flow diagram of the SA optimization is described and shown in Fig. 4.
6. RESULTS AND DISCUSSION
6.1. ResultsThe accuracy of the SA optimization depends on the cooling rate (kk) and the number of
iterations (Iter). To achieve good optimization, both the cooling rate (kk) and the number ofiterations (Iter) are varied step by step
Fig. 4. Flow diagram of a SA optimization.
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kk ~ 0:90,0:93,0:96,0:99ð Þ; Iter ~ 50,200,800ð Þ
The results of two kinds of optimizations — one of the pure tone noise and the others of thebroadband noise — are described below.
Pure Tone Noise Optimization:Six sets of SA parameters are tested by varying the values of the SA parameters. The
simulated results with respect to the pure tone of 300 Hz is summarized and shown inTable 3. As indicated in Table 3, the optimal design data can be obtained from the last setof SA parameters at (kk, Iter) 5 (0.99, 800). Using the optimal design in a theoreticalcalculation, the optimal STL curves with respect to various SA parameters are plotted anddepicted in Fig. 5. As revealed in Fig. 5, the optimal STL is maximized at the desiredfrequency.
Table 3. Optimal STL for a muffler with reverse-flow ducts (at a targeted tone of 300 Hz).
SA parameters Results
kk Iter
0.90 50 RT1 RT2 RT3 RT4 STL (dB)0.8997 0.8993 0.3997 0.3997 19.5g1 dh1(m) g2 dh2(m)0.006996 0.09994 0.006996 0.09994
0.93 50 RT1 RT2 RT3 RT4 STL (dB)0.8785 0.8570 0.3839 0.3839 24.5g1 dh1(m) g2 dh2(m)0.006718 0.09623 0.006718 0.09623
0.96 50 RT1 RT2 RT3 RT4 STL (dB)0.8657 0.8315 0.3743 0.3743 26.0g1 dh1(m) g2 dh2(m)0.006550 0.09400 0.006550 0.09400
0.99 50 RT1 RT2 RT3 RT4 STL (dB)0.8405 0.7810 0.3554 0.3554 26.5g1 dh1(m) g2 dh2(m)0.006219 0.08959 0.006219 0.08959
0.99 200 RT1 RT2 RT3 RT4 STL (dB)0.8241 0.7482 0.3431 0.3431 27.5g1 dh1(m) g2 dh2(m)0.006004 0.08672 0.006004 0.08672
0.99 800 RT1 RT2 RT3 RT4 STL (dB)0.6875 0.4749 0.2406 0.2406 85.5g1 dh1(m) g2 dh2(m)0.004210 0.06281 0.004210 0.06281
Notes: RT15 LZ/Lo; RT25 LC/LZ; RT 35 D1/Do; RT45 D2/Do; Lo5L1+LZ; LZ5 LA+LB+LC; LA5LB5
(LZ-LC)/2.
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Broadband Noise Optimization:By using the above SA parameters, the muffler’s optimal design data for mufflers hybridized
with reverse-flow perforated ducts used to minimize the sound power level at the muffler’soutlet is summarized in Table 4. As illustrated in Table 4, the resultant sound power levels havebeen dramatically reduced from 140.7 dB to 42.5 dB. Using this optimal design in a theoreticalcalculation, the resultant SWL before and after adding the muffler at the outlet is shown inFig. 6. As shown in Fig. 6, the muffler has the best acoustical performance.
6.2. DiscussionTo achieve a sufficient optimization, the selection of the appropriate SA parameter set is
essential. As indicated in Table 3, the best SA set at the targeted pure tone noise of 300 Hz has
Fig. 5. The STL with respect to frequencies at various SA parameters (targeted tone: 300Hz).
Table 4. Optimal SWL for a muffler with reverse-flow ducts (broadband noise).
ItemSA parameters Results
kk Iter
1 0.99 50 RT1 RT2 RT3 RT4 SWLT (dB)0.8227 0.7453 0.3420 0.3420 75.5g1 dh1(m) g2 dh2(m)0.005985 0.08647 0.005985 0.08647
2 0.99 200 RT1 RT2 RT3 RT4 SWLT (dB)0.8329 0.7657 0.3496 0.3496 71.9g1 dh1(m) g2 dh2(m)0.006119 0.08825 0.006119 0.08825
3 0.99 800 RT1 RT2 RT3 RT4 SWLT (dB)0.7007 0.5015 0.2506 0.2506 42.5g1 dh1(m) g2 dh2(m)0.004385 0.06513 0.004385 0.06513
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been shown. Using the appropriate SA set at the targeted pure tone (300 Hz), the related optimalSTL curves plotted in Fig. 5 reveals that the predicted maximal value of the STL is located at thedesired frequency. Therefore, using the SA optimization in finding a better design solution hasproven reliable; moreover, as can be seen in Fig. 5, not only the peak frequency of the STL profileis shifting toward the targeted tone but the peak value will increase when a better solution assessedin the SA optimization procedure. Meanwhile, all of the design data (RT1, RT2, RT3, RT4, g1, dh1,
g2, dh2) were decrease. It means that the shorter of the length of the perforated tube and thesmaller diameter and number of the perforated hole will result in a higher frequency. Besides, theSTL will also increase when the diameter of the perforated tubes decrease.
7. CONCLUSION
It has been shown that mufflers hybridized with reversed-flow and perforated ducts can beeasily and efficiently optimized within a limited space by using a generalized decouplingtechnique, a plane wave theory, a four-pole transfer matrix, as well as a SA optimizer. Two kindsof SA parameters (kk, Iter) play essential roles in the solution’s accuracy during the SA
optimization. As indicated in Fig. 5, the tuning ability established by adjusting design of themufflers is reliable. In additional, the design data reveals that the peak frequency of the STL willshift rightward if the length of the perforated tube, the diameter of the perforated hole, and thenumber of the perforated hole decrease. Besides, the whole STL will increase if the diameter of theperforated tubes decreases. Subsequently, the appropriate acoustical performance curve ofmufflers with reverse-flow and perforated ducts in depressing overall broadband noise have beenassessed. As can be seen in Table 4 and Fig. 6, the overall sound power level (SWLT) of a blower isminimized by adjusting an appropriate spectrum of the STL. Consequently, the approach usedfor the optimal design of the STL proposed in this study is indeed easy and quite effective.
ACKNOWLEDGEMENTS
The author acknowledges the financial support of the National Science Council (NSC 97-2622-E-235-002-CC3, Taiwan, ROC) and would also like to thank the anonymous referees whokindly provided the suggestions and comments to improve this work.
Fig. 6. Optimal STL for mufflers designed at various SA parameters (broadband noise).
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REFERENCES
1. Davis, D.D., Stokes, J.M., Moorse, L., ‘‘Theoretical and experimental investigation of mufflers
with components on engine muffler design,’’ NACA Report, pp.1192, 1954.
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cavities,’’ Acoustical Society of America, Vol. 64, pp. 207–215, 1978.
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Acoustical Society of America, Vol. 66, pp. 772–778, 1979.
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Acoustical Society of America, Vol. 66, pp. 779–788, 1979.
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Transactions of the Canadian Society for Mechanical Engineering, Vol. 34, No. 1, 2010 28
APPENDIX A-TRANSFER MATRIX OF A REVERSE-FLOW PERFORATED DUCT
As shown in Fig. 7, there are six nodes located inside the acoustical field. Based on thederivation from Munjal et al. [14], the continuity equations and momentum equations withrespect to the inner and outer tubes in the first chamber are listed below.
Inner tube 1:
continuity equation
V2Lr2
Lxzro
Lu2
Lxz
4ro
D1u2,3z
Lr2
Lt~0 ðA1Þ
momentum equation
ro
LLt
zV2LLx
� �u2z
Lp2
Lx~0 ðA2Þ
Inner tube 2:
continuity equation
V4Lr4
Lxzro
Lu4
Lx{
4ro
D8u3,4z
Lr4
Lt~0 ðA3Þ
momentum equation
ro
LLt
zV4LLx
� �u4z
Lp4
Lx~0 ðA4Þ
Outer tube:
continuity equation
ro
Lu3
Lx{V3
Lr3
Lx{
4D1ro
D2o{D2
1{D22
u2,3z4D2
D2o{D2
1{D22
rou2,3zLr3
Lt~0 ðA5Þ
momentum equation
Fig. 7. The mechanism of an acoustical element for a muffler with reverse -flow perforated ducts.
Transactions of the Canadian Society for Mechanical Engineering, Vol. 34, No. 1, 2010 29
ro
LLt
zV3LLx
� �u3z
Lr3
Lx~0 ðA6Þ
Assuming that the acoustic wave is a harmonic motion
p x,tð Þ~P xð Þ:ejvt ðA7Þ
under the isentropic processes in ducts, it yields
P xð Þ~r xð Þ:c2o ðA8Þ
Assuming that the perforation along the inner tubes is uniform (ie.dj=dx~0), the acousticimpedance of the perforation (rocoj) is
rocoj1~p2 xð Þ{p3 xð Þ
u2,3 xð Þ ðA9Þ
rocoj2~p2 xð Þ{p3 xð Þ
u2,3 xð Þ ðA10Þ
where j1,j2 are the specific acoustical impedances of the inner perforated tube1 andtube 2, respectively. According to the formula of j developed by Sullivan [3] and Rao [17],the empirical formulations for the perforation with or without mean flow are adopted in thisstudy.
For perforates with stationary medium, we have
j1~ 0:006zjk t1z0:75dh1ð Þ½ �=g1 ðA11aÞ
j2~ 0:006zjk t2z0:75dh2ð Þ½ �=g2 ðA11bÞ
For perforates with grazing flow, we have
j1~ 0:514D1M2= LCg1ð Þzj0:95k t1z0:75dh1ð Þ½ �=g1 ðA12aÞ
j2~ 0:514D3M4= LCg2ð Þzj0:95k t2z0:75dh2ð Þ½ �=g2 ðA12bÞ
where dh1 and dh2 are the diameters of the perforated holes on the inner tube 1 and the tube 2; t1
and t2 are the thicknesses of the inner perforated tube 1 and tube 2; g1 and g2 are the porositiesof the perforated tube 1 and tube 2.
Transactions of the Canadian Society for Mechanical Engineering, Vol. 34, No. 1, 2010 30
The available ranges of the above parameters are [17]
M : 0:05 ¼v M2, M4 ¼v 0:2 ðA13aÞ
g : 0:03 ¼v g1, g2 ¼v 0:1 ðA13bÞ
t : 0:001 ¼v t1, t2 ¼v 0:003 ðA13cÞ
dh : 0:00175 ¼v dh1, dh2 ¼v 0:007 ð13dÞ
Eliminating u2,u4,u2,3,u3,4,r2,r3 and r4 yields
D2za1Dza2 a3Dza4 0
a5Dza6 D2za7Dza8 a9Dza10
0 a11Dza12 D2za13Dza14
264
375
p2 xð Þp3 xð Þp4 xð Þ
264
375~
0
0
0
264375 ðA14Þ
Developing Eq.(A14) yields
p00
2za1p0
2za2p2za3p0
3za4p3~0 ðA15aÞ
a5p0
2za6p2zp00
3za7p0
3za8p3za9p0
4za10p4~0 ðA15bÞ
a11p0
3za12p3zp00
4za13p0
4za14p4~0 ðA15cÞ
where
D~d
dx; a1~{
jM2
1{M22
2k{j4
D1j1
� �; a2~
1
1{M22
k2{j4k
D1j1
� �;
a3~M2
1{M22
: 4
D1j1
; a4~{j
1{M22
: 4k
D1j1
; a5~M3
1{M23
: 4D1
(D2o{D2
1{D22)j1
;
a6~j
1{M23
: 4kD1
(D2o{D2
1{D22)j1
; a7~{jM3
1{M23
2k{j4D1
(D2o{D2
1{D22)j1
{j4D2
(D2o{D2
1{D22)j2
� �;
a8~1
1{M3k2{
j4kD1
(D2o{D2
1{D22)j1
{j4kD2
(D2o{D2
1{D22)j2
� �;
a9~M3
1{M23
4D2
(D2o{D2
1{D22)j2
� �; a10~
j
1{M23
4kD2
(D2o{D2
1{D22)j2
� �;
a11~M4
1{M24
4
D2j2
� �; a12~
j
1{M24
4k
D2j2
� �; a13~
{jM4
1{M24
2k{j4
D2j2
� �;
a14~1
1{M24
k2{j4k
D2j2
� �;
ðA15dÞ
Transactions of the Canadian Society for Mechanical Engineering, Vol. 34, No. 1, 2010 31
Let p0
2~dp2
dx~y1, p
0
3~dp3
dx~y2, p
0
4~dp4
dx~y3, p2~y4, p3~y5, p4~y6
The new matrix between {y’} and {y} is
y0
1
y0
2
y0
3
y0
4
y05
y0
6
2666666664
3777777775~
{a1 {a3 0 {a2 {a4 0
{a5 {a7 {a9 {a6 {a8 {a10
0 {a11 {a13 0 {a12 {a14
1 0 0 0 0 0
0 1 0 0 0 0
0 0 1 0 0 0
2666666664
3777777775
y1
y2
y3
y4
y5
y6
2666666664
3777777775
ðA16aÞ
which can be briefly expressed as
y0
n o~ LL½ � yf g ðA16bÞ
Let
yf g~
dp2=dx
dp3=dx
dp4=dx
p2
p3
p4
2666666664
3777777775~
PP1,1 PP1,2 PP1,3 PP1,4 PP1,5 PP1:6
PP2,1 PP2,2 PP2,3 PP2,4 PP2,5 PP2,6
PP3,1 PP3,2 PP3,3 PP3,4 PP3,5 PP3,6
PP4,1 PP4,2 PP4,3 PP4,4 PP4,5 PP4,6
PP5,1 PP5,2 PP5,3 PP5,4 PP5,5 PP5,6
PP6,1 PP6,2 PP6,3 PP6,4 PP6,5 PP6,6
2666666664
3777777775
CC1
CC2
CC3
CC4
CC5
CC6
2666666664
3777777775
ðA17Þ
where PP½ �6x6 is the modal matrix formed by six sets of eigen vectors PP6x1 of LL½ �6x6.
Combining Eq.(A17) with (A16) and then multiplying PP½ �{1by both sides, we have
PP½ �{1 PP½ � CC0
n o~ PP½ �{1 LL½ � PP½ � CCf g ðA18Þ
Set VV½ �~ PP½ �{1 LL½ � PP½ �~
l1 0 0 0 0 0
0 l2 0 0 0 0
0 0 l3 0 0 0
0 0 0 l4 0 0
0 0 0 0 l5 0
0 0 0 0 0 l6
2666666664
3777777775
ðA19Þ
where li is the eigen value of LL½ �.
Transactions of the Canadian Society for Mechanical Engineering, Vol. 34, No. 1, 2010 32
Eq.(A17) can be thus rewritten as
CC0
n o~ VV½ � CCf g ðA20Þ
Obviously, Eq.(20) is a decoupled equation. The related solution yields
CCi~kfielix ðA21Þ
Using Eqs.(A2),(A4),(A6),(A17) and (A21), the relationship of the acoustic pressure and theparticle velocity yields
p2(x)
p3(x)
p4(x)
rocou2(x)
rocou3(x)
rocou4(x)
2666666664
3777777775~
E1,1 E1,2 E1,3 E1,4 E1,5 E1,6
E2,1 E2,2 E2,3 E2,4 E2,5 E2,6
E3,1 E3,2 E3,3 E3,4 E3,5 E3,6
E4,1 E4,2 E4,3 E4,4 E4,5 E4,6
E5,1 E5,2 E5,3 E5,4 E5,5 E5,6
E6,1 E6,2 E6,3 E6,4 E6,5 E6,6
2666666664
3777777775
kf1
kf2
kf3
kf4
kf5
kf6
2666666664
3777777775
ðA22aÞ
where
E1,i~PP4,ielix; E2,i~PP5,ie
lix; E3,i~PP6,ielix; E4,i~{
PP1,ielix
jkzM2li
;
E5,i~{PP2,ie
lix
jkzM3li
; E6,i~{PP3,ie
lix
jkzM4li
ðA22bÞ
Plugging x50 and x5Lc into Eq.(A22) and doing rearrangement yield
p2 0ð Þp3 0ð Þp4 0ð Þ
rocou2 0ð Þrocou3 0ð Þrocou4 0ð Þ
2666666664
3777777775~ Y½ �
p2 LCð Þp3 LCð Þp4 LCð Þ
rocou2 LCð Þrocou3 LCð Þrocou4 LCð Þ
2666666664
3777777775
ðA23aÞ
Transactions of the Canadian Society for Mechanical Engineering, Vol. 34, No. 1, 2010 33
where
Y½ �~ E 0ð Þ½ � E LCð Þ½ �{1 ðA23bÞ
To obtain the transform matrix between the inlet (x50) and the outlet (x5Lc) of the innertubes, four boundary conditions for the outer tube at x50 and x5Lc are placed in thecalculation.
p3 0ð Þ{u3 0ð Þ~{jroco cot kLAð Þ ðA24aÞ
p2 LCð Þu2 LCð Þ~{jroco cot kLBð Þ ðA24bÞ
p3 LCð Þu3 LCð Þ~{jroco cot kLBð Þ ðA24cÞ
p4 LCð Þu4 LCð Þ~{jroco cot kLBð Þ ðA24dÞ
By combining Eqs.(A23a, b, c, d) with Eq.(A24) and developing them, the transfer matrixyields
p2 0ð Þrocou2 0ð Þ
� �~
TPRF21,1 TPRF21,2
TPRF22,1 TPRF22,2
� �p4 LCð Þ
rocou4 LCð Þ
� �ðA25aÞ
or in a brief form
�pp2
roco�uu2
� �~
TPRF21,1 TPRF21,2
TPRF22,1 TPRF22,2
� ��pp4
roco�uu4
� �ðA25bÞ
Transactions of the Canadian Society for Mechanical Engineering, Vol. 34, No. 1, 2010 34
where�pp2~p2(0); �uu2~u2(0); �pp4~p4(LC); �uu4~{u4(LC);
TPRF21,1~H15
H17; TPRF21,2~
1
roco
: H15H18
H17{H16
� �; TPRF22,1~
roco
H17;
TPRF22,2~H18
H19;
K11~roco Y14{jY11 cot kLBð Þ½ �; K12~roco Y15{jY12 cot kLBð Þ½ �;
K13~roco Y16{jY13 cot kLBð Þ½ �; K21~roco Y24{jY21 cot kLBð Þ½ �;
K22~roco Y25{jY22 cot kLBð Þ½ �; K23~roco Y26{jY23 cot kLBð Þ½ �;
K31~roco Y34{jY31 cot kLBð Þ½ �; K32~roco Y35{jY32 cot kLBð Þ½ �;
K33~roco Y36{jY33 cot kLBð Þ½ �; K41~roco Y44{jY41 cot kLBð Þ½ �;
K42~roco Y45{jY42 cot kLBð Þ½ �; K43~roco Y46{jY43 cot kLBð Þ½ �;
K51~roco Y54{jY51 cot kLBð Þ½ �; K52~roco Y55{jY52 cot kLBð Þ½ �;
K53~roco Y56{jY53 cot kLBð Þ½ �; K61~roco Y64{jY61 cot kLBð Þ½ �;
K62~roco Y65{jY62 cot(kLB)½ �; K63~roco Y66{jY63 cot kLBð Þ½ �;
H1~j cot kLAð Þ:K52{K22
K21{jK51 cot kLAð Þ ; H2~j cot kLAð Þ:K53{K23
K21{jK51 cot kLAð Þ ; H3~K11H1zK12;
H4~K11H2zK13; H5~K31H1zK32; H6~K31H2zK33; H7~K41H1zK42;
H8~K41H2zK43; H9~K61H1zK62; H10~K61H2zK63; H11~rocoH10
H7H10{H8H9;
H12~{rocoH8
H7H10{H8H9; H13~
rocoH9
H8H9{H7H10; H14~
{rocoH7
H8H9{H7H10;
H15~H3H11zH4H13; H16~H3H12zH4H14; H17~H5H11zH6H13;
H18~H5H12zH6H14
ðA25cÞ
Transactions of the Canadian Society for Mechanical Engineering, Vol. 34, No. 1, 2010 35
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